Phantom maps and chromatic phantom maps
by J. Daniel Christensen and Mark Hovey

In this paper, we introduce n-phantom maps.  These are maps
f: X--> Y such that for all finite spectra W of type at least n and all
maps g: W--> X, the composite fg is trivial.  We prove a number of
properties of n-phantom maps analogous to properties of phantom maps.
One advantage of n-phantom maps is that the kernel on homotopy of the
map L_n S^0 --> L_K(n) S^0 consists precisely of the n-phantom homotopy
classes, and there are no n-phantom maps to L_K(n) S^0.